What did the Fire bring to the market that didn't already exist?
They can offer a cheapan inexpensive device, with deep integration to Amazon's ecosystem.
Maybe they tie the smartphone contact into Prime. Their ultimate aim seems to be to have everybody be a Prime member for one reason or another.

Cheap wasn't mean to have a negative tone, although I can see that sometimes it can be taken that way.
I would say that compared to the competition, both the Nexus 4 and 7 are cheap, but they both seem like very good devices. (For the price, I'd say the 7 was a great device)

Aging baby boomers are an obvious growing market - create a smart phone for the non-tech savvy population - like the Fire put an Amazon "skin" on an Android tablet so folks can use it without knowing anything about Android - put an Amazon "skin" on a smart phone.

Why? It is unlikely that Amazon will be pitching this as a high-end phone to compete with the iPhone and Galaxy SIII, it will more likely be a mid-level phone designed to push Amazon's ecosystem, as with the Fire. Its direct competition will be something like the Nexus 4.
It is more threatening to Google, from that point of view, as it increases the marketshare of the proprietary Android systems, versus the more open ones. Even more devices with custom UIs that Google didn't design, even more devices that don't get timely updates, even more devices with non-standard app stores that Google doesn't get revenue from, and (the greatest threat), more chance that at some point Amazon introduce their own in-app ad system, and require developers to use it. It was that risk with Apple that was a significant motivation for developing Android in the first place, Google can't afford to be locked out of ad revenue.

Jeff Bezos said the Fire was not competing with the iPad. It wasn't. Amazon is competing with Apple. It's a clash of the ecosystems. Apple knows this and that is why they scrambled to get the mini to market. Apple's market share has dropped dramatically. Developers are flirting with other platforms. It's a big deal. When Amazon introduces their phone, it won't compete with the iPhone, but Amazon will be more competitive with Apple. Apple has to grow a lot to sustain their stock price and now they have to be careful about price since Amazon offers something pretty similar for a lot less money.

Well when they introduced the iPad, they had essentially 100% of the market. Share can only go down from there
There will obviously be a knock-on effect, but the immediate conflict will be between Amazon and Google rather than Amazon and Apple. The Nexus 4 already offers a high quality phone for much lower prices.
In the same way that the Nexus 7 was primarily designed to compete with the 'impure' Android devices from Amazon and B&N, rather than the iPad.

Then the mini, like most Apple products, is stuck in it's own little world.

And the mini's not a phone, so really has nothing to say on the matter.

Also the mini is priced way above where Google and Amazon prices their stuff.

The Google and Amazon offerings will likely be phones that are priced similarly, running more or less the same operating system and the same apps, and targeted at the same market segment.

So the mini is entitled to it's opinion, but it's wrong.
Even if it were the iPhone 5 speaking, rather than the iPad mini, it would still be wrong, for all the same reasons (except the "not a phone" one).